For any two paths \(\xi _j\in {\cal P}_{\tau }(2n_j)\) 
with \(j=0\)  and 1, let \(\xi _0{\rm \diamond }\xi _1(t)= \xi _0(t){\rm \diamond }\xi _1(t)\)  for all \(t\in [0,\tau ]\) .
As in [1]}, for \({\lambda }\in {\bf R}\setminus \lbrace 0\rbrace \) , \(a\in {\bf R}\) , \({\theta }\in (0,\pi )\cup (\pi ,2\pi )\) ,
\(b=({\begin{matrix}b_1 & b_2\\b_3 & b_4\end{matrix}})\)  with \(b_i\in {\bf R}\)  for \(i=1, \ldots , 4\) , and \(c_j\in {\bf R}\) 
for \(j=1, 2\) , some normal forms are given by
\(& D({\lambda })=\begin{pmatrix}{\lambda }& 0\\0 & {\lambda }^{-1} \end{pmatrix}, \qquad R({\theta })=\begin{pmatrix}\cos {\theta }& -\sin {\theta }\\\sin {\theta }& \cos {\theta }\end{pmatrix}, \qquad N_1({\lambda }, a)=\begin{pmatrix}{\lambda }& a\\0 & {\lambda }\end{pmatrix},\\& N_2(e^{\sqrt{-1}{\theta }},b) = \begin{pmatrix}R({\theta }) & b\\0 & R({\theta })\end{pmatrix}, \quad M_2({\lambda },c)=\begin{pmatrix}{\lambda }& 1 & c_1 & 0 \\0 & {\lambda }& c_2 & (-{\lambda })c_2 \\0 & 0 & {\lambda }^{-1} & 0 \\0 & 0 & -{\lambda }^{-2} & {\lambda }^{-1} \end{pmatrix}. \) 
